How is weighted average computed in Deep Q networks
Let's say $Q$ is the old estimate, $Q'$ the new estimate, and $R$ is the return.
We have
$$ 
Q' = Q + \alpha(R-Q)
$$
This can be rewritten as
$$ 
Q' = (1-\alpha)Q + \alpha R
$$
When $\alpha$ is a constant, this is an exponential weighted average of returns. If $n$ is the number of samples we get and $\alpha=1/n$ (so it decreases with each sample), we get
$$
Q' = \frac{n-1}{n}Q + \frac{1}{n}R
$$
This simply represents the average return. So, playing with $\alpha$ tunes the weighting of the estimate.